Unsupervised Approach Data Analysis Based on Fuzzy Possibilistic Clustering: Application to Medical Image MRI

Abstract

The analysis and processing of large data are a challenge for researchers. Several approaches have been used to model these complex data, and they are based on some mathematical theories: fuzzy, probabilistic, possibilistic, and evidence theories. In this work, we propose a new unsupervised classification approach that combines the fuzzy and possibilistic theories; our purpose is to overcome the problems of uncertain data in complex systems. We used the membership function of fuzzy c-means (FCM) to initialize the parameters of possibilistic c-means (PCM), in order to solve the problem of coinciding clusters that are generated by PCM and also overcome the weakness of FCM to noise. To validate our approach, we used several validity indexes and we compared them with other conventional classification algorithms: fuzzy c-means, possibilistic c-means, and possibilistic fuzzy c-means. The experiments were realized on different synthetics data sets and real brain MR images.

1. Introduction

Image segmentation is a very important operation in the process of treatment and analyzing images, and it is widely used in the different fields: pattern recognition, remote sensing, artificial intelligence, medical imaging, and so forth. The field of medical imaging includes several types of images: radiography (X-ray), ultrasound and magnetic resonance image [1–4]. These images are a very complex data, so their analysis is a challenge for researches.

In the literature, there are several methods that can segment these images. We can group them in four classes. The first one is the Thresholding; it allows to find the optimal threshold value, in order to extract the background objects in the image. In general, this approach is very sensitive to noise and ignores the spatial parameters [5, 6].

The second approach is the Contour; it allows to detect the contour of the image. This method is easy to implement, but unfortunately it is very sensitive to the noise and also to the parameters initialization, which means that it is mostly used with a pretreatment filter [7–10].

The third approach is the Region, which generates some methods: growing region (called ascendant) and splitting/merging (called descendants); this approach is very sensitive to the initial parameters and to the noise [11–13].

The last approach is the Clustering; it is a very important operation in the process and data analysis, and it allows creating the homogeneous partitions using a similarity criterion [3, 4, 14–33].

In this work, we are interested in clustering segmentation using the possibility theory combined with fuzzy theory.

The remainder of this paper is structured as follows. In Section 2, we present the clustering approach with three conventional algorithms, fuzzy c-means (FCM) (Algorithm 1), possibilistic c-means (PCM) (Algorithm 2), and possibilistic fuzzy c-means (PFCM) (Algorithm 3). In Section 3, the proposed new approach (HPFCM) is formulated to cluster the complex data set. In Section 4, we present the experimental results using different synthetics data sets and real MR images. Finally, the conclusion and perspectives are presented in Section 5.

Let 𝑋={𝑋𝑖/𝑖={1⋯𝑁}} the data sets, 𝐾: the number of cluster, 𝑈 = (𝑢𝑖𝑗)1≤𝑖≤𝑁,1≤𝑗≤𝐾 is the matrix of membership degrees function, 𝑉=(𝑣𝑖𝑗)1≤𝑖≤𝑁,1≤𝑗≤𝐾 is the matrix of cluster centers, 𝑚: the degree of fuzzy, and 𝜀: is the threshold representing convergence error. (1) Step 1. (2)  (i) Initialize the matrix 𝑈 by random values in the interval [0,1] and it also satisfies the condition in (3). (3) Step 2. (4)   Repeat (5)  (i) Update the matrix 𝑉 cluster centers in (5). (6)  (ii) Update the matrix 𝑈 degree of membership in (4). (7)   Until (8)    Stability of the matrix 𝑉,  𝑉new−𝑉old≤𝜀

Algorithm 1 

FCM.

Let 𝑋={𝑋𝑖/𝑖={1⋯𝑁}} the data sets, 𝐾: the number of cluster, 𝑈=(𝑢𝑖𝑗)1≤𝑖≤𝑁,1≤𝑗≤𝐾 is the matrix of membership degrees function, 𝑉=(𝑣𝑖𝑗)1≤𝑖≤𝑁,1≤𝑗≤𝐾 is the matrix of cluster centers, 𝑚: the degree of fuzzy, 𝜂𝑗: Degree of weight possibilistic, and 𝜀: is the threshold representing the convergence error.  Step 1. (2)  (i) Initialize the matrix 𝑈 by random values in the interval [0,1].   Step 2. (4)   Repeat    (i) Update the matrix 𝑉 cluster centers in (8). (6) (ii) Update the matrix 𝑈 degree of membership in (7).     Until (8)  Stability of the matrix 𝑉,𝑉new−𝑉old≤𝜀

Algorithm 2 

PCM.

Let 𝑋={𝑋𝑖/𝑖={1⋯𝑁}} the data sets, 𝐾: the number of cluster, 𝑈 = (𝑢𝑖𝑗)1≤𝑖≤𝑁,1≤𝑗≤𝐾 is the matrix of membership degrees function, 𝑉=(𝑣𝑖𝑗)1≤𝑖≤𝑁,1≤𝑗≤𝐾 is the matrix of cluster centers, 𝑚: the degree of fuzzy, 𝜂𝑗: Degree of weight possibilistic, and 𝜀: is the threshold representing the convergence error.   Step  1. (2)  (i) Initialize the matrix 𝑈 by random values in the interval [0,1].   Step  2. (4)   Repeat    (i) Update the matrix 𝑉 cluster centers in (12). (6)   (ii) Update the matrix 𝑈 degree of membership in (10).   Until (8)   Stability of the matrix 𝑉,𝑉new−𝑉old≤𝜀

Algorithm 3 

PFCM.

2. Theories of Data Analysis

2.1. Classification Approach

Classification is a method that allows analysing and treating the complex data. It is widely used in different fields: pattern recognition, remote sensing, images processing, and artificial intelligence [3, 14, 16].

In the literature there are two types of classification: supervised and unsupervised. In general, the supervised approach uses a learning base in order to extract and produce the decision functions for classifying data. But, the unsupervised classification is used without a learning knowledge base, also called clustering [17, 22, 34, 35].

In this work, we are interested in studying clustering approach and its applications in medical image processing.

2.2. Clustering

Clustering is a process that allows dividing data into groups of similar pattern, and these groups are called clusters. In the literature, there are several theories of clustering. The first notion of fuzzy theory was proposed by Zadeh [36], and he established the basic principle of fuzzy theory using the fuzzy logic, in order to describe the uncertainty of belonging by a membership function. Then Ruspini [35] proposed the first notion of a fuzzy partition, and he considers that each cluster is a fuzzy set. Zadeh [34] proposed the conceptual framework for cluster analysis and pattern classification using the fuzzy set theory. Later, several researches were published; in order to improve the Bezdek algorithm, Rousseeuw et al. [37–39] proposed the different objective functions that allow improving the effectiveness of fuzzy c-means algorithm.

To overcome the weakness of FCM to noise, Krishnapuram and Keller [23] proposed to relax the constraint of fuzzy, and they established the first possibilistic c-means algorithm (PCM). Then Barni et al. [40] showed that PCM algorithm is very sensitive to initializations and generates the coincident clusters. Timm et al. [29] proposed the possibilistic fuzzy clustering, and Pal et al. [30] proposed another possibilistic fuzzy c-means (PFCM) that can avoid the coincident cluster of PCM and be less sensitive to the noise.

2.3. Fuzzy c-Means

Fuzzy c-means algorithm was established by Bezdek [22], it allows classifying uncertain and imprecise data, and it is the most widely used in fuzzy clustering. The FCM model is the optimization problem 𝐽_𝑚: 𝐽_𝑚(𝑈,𝑉;𝑋)=^𝑁𝐾_𝑖=1𝑗=1𝑢^𝑚_𝑖𝑗𝑥_𝑖−𝑣_𝑗²_𝐴, (1) where 𝑋={𝑋_𝑖,𝑖=1⋯𝑁}⊆ℜ^𝑞 is the data set, 𝑛 is the number of data, 𝐾 is the number of clusters, 𝑚 is the degree of fuzzy, 𝑢_𝑖𝑗 is the degree of membership, 𝑣_𝑗 is the center of cluster 𝑗, and ‖𝑥_𝑖−𝑣_𝑗‖_𝐴 is a distance between 𝑉_𝑗 and the object 𝑥_𝑖. Consider Ψ_𝐾𝑁=𝑈∈ℜ^𝑁𝐾∶0≤𝑢_𝑖𝑗≤1,∀𝑖,∀𝑗;∀𝑖∃𝑗𝑢_𝑖𝑗,𝑀>0_𝑓𝑐=𝑈∈Ψ_𝐾𝑁∶^𝐾_𝑗=1𝑢_𝑖𝑗=1,∀𝑖∈{1,…,𝑁_𝐾𝑖=1};𝑢_𝑖𝑗>0,∀𝑗∈{1,…,𝑁}. (2)(3)

Theorem 1 (FCM). If 𝐷_𝑖𝑗𝐴=‖𝑥_𝑖−𝑣_𝑗‖_𝐴>0, for all 𝑖, 𝑗, 𝑚>1, and 𝑋 data set contains at least 𝐾 different patterns, then (𝑈,𝑉)∈𝑀_𝑓𝑐×ℜ^𝐾×𝑞 and 𝐽_𝑚 can be minimized only if 𝑢_𝑖𝑗=^𝐾_𝑠=1𝑥_𝑖−𝑣_𝑗²_𝐴𝑥_𝑠−𝑣_𝑗²_𝐴^{1/(𝑚−1)−1},{}{},𝑣𝑖∈1,…,𝑁,𝑗∈1,…,𝐾_𝑗=∑^𝑁_𝑖=1𝑢^𝑚_𝑖𝑗𝑥_𝑖∑^𝑁_𝑖=1𝑢^𝑚_𝑖𝑗,∀𝑗∈{1,…,𝐾}. (4)(5)

2.4. Possibilistic c-Means

Possibilistic c-means algorithm was introduced by Krishnapuram and Keller [23] to overcome the sensitivity of fuzzy c-means to noise. Their idea allows relaxing the fuzzy constraint.

The PCM is the optimization problem 𝐽_pcm: 𝐽_pcm=𝑈,𝑉;𝑋,𝜂_{𝑁𝐾𝑖=1𝑗=1}𝑢^𝑚_𝑖𝑗𝑥_𝑖−𝑣_𝑗²_𝐴+^𝐾_𝑗=1𝜂_𝑗^𝑁_𝑗=11−𝑢_𝑖𝑗^𝑚,0≤𝑢_𝑖𝑗≤1,∀𝑖∈{1,…,𝑁},∀𝑗∈{1,…,𝐾}. (6)

Theorem 2 (PCM). If 𝐷_𝑖𝑗𝐴=‖𝑥_𝑖−𝑣_𝑗‖_𝐴>0, for all 𝑖, 𝑗, 𝑚>1, 𝜂>1, and 𝑋 data set contains at least 𝐾 different patterns, then (𝑈,𝑉)∈Ψ_𝐾𝑁×ℜ^𝐾×𝑞 and 𝐽_𝑝𝑐𝑚 can be minimized only if 𝑢_𝑖𝑗=1+‖𝑥_𝑖−𝑣_𝑗‖²_𝐴𝜂_𝑗^{−1/(𝑚−1)},𝑣1≤𝑖≤𝑁,1≤𝑗≤𝐾,_𝑗=∑^𝑁_𝑖=1𝑢^𝑚_𝑖𝑗𝑥_𝑖∑^𝑁_𝑖=1𝑢^𝑚_𝑖𝑗,∀𝑗∈{1,…,𝐾}. (7)(8)

2.5. Possibilistic Fuzzy c-Means (PFCM)

Pal et al. [30] proposed the possibilistic fuzzy c-means algorithm (PFCM) that is based on FCM and PCM, and it generates the two membership functions concurrently; the first is a possibilistic membership (𝑡_𝑖𝑗) that presents the absolute degree of typicality, and the second function is the fuzzy membership (𝑢_𝑖𝑗) that presents the relative degree.

The PFCM is the optimization problem 𝐽_pfcm: 𝐽_pfcm=𝑈,𝑉;𝑋,𝜂_{𝑁𝐾𝑖=1𝑗=1}𝑎𝑢^𝑚_𝑖𝑗+𝑏𝑡^𝜂_𝑖𝑗𝑥_𝑖−𝑣_𝑗²_𝐴+^𝐾_𝑗=1𝛾_𝑗^𝑁_𝑗=1(1−𝑡_𝑖𝑗)^𝜂, (9) where 𝑎>0,𝑏>0,𝑚>1, and 𝜂>1.

Theorem 3 (PFCM). If 𝐷_𝑖𝑗𝐴=‖𝑥_𝑖−𝑣_𝑗‖_𝐴>0, for all 𝑖, 𝑗, 𝑚>1, 𝜂>1, and 𝑋 data set contains at least 𝐾 different patterns, then (𝑈,𝑇,𝑉)∈𝑀_𝑓𝑐×Ψ_𝐾𝑁×ℜ^𝐾×𝑞 and 𝐽_{𝑝𝑓𝑐𝑚} can be minimized only if 𝑢_𝑖𝑗=^𝐾_𝑠=1𝑥_𝑖−𝑣_𝑗²_𝐴𝑥_𝑠−𝑣_𝑗²_𝐴^{1/(𝑚−1)−1},𝑡1≤𝑖≤𝑁,1≤𝑗≤𝐾,_𝑖𝑗=11+𝑏/𝛾_𝑗𝑥_𝑖−𝑣_𝑗²_𝐴^1/(𝜂−1),𝑣1≤𝑖≤𝑁,1≤𝑗≤𝐾,_𝑗=∑^𝑁_𝑖=1𝑎𝑢^𝑚_𝑖𝑗+𝑏𝑡^𝜂_𝑖𝑗𝑥_𝑖∑^𝑁_𝑖=1𝑎𝑢^𝑚_𝑖𝑗+𝑏𝑡^𝜂_𝑖𝑗,1≤𝑗≤𝐾. (10)(11)(12)

Pal et al. [30] showed that PFCM can overcome the weakness of FCM to noise and also overcome the problem of coincident class of PCM.

3. The Proposed Algorithm HPFCM

3.1. HPFCM

Our new hybrid approach HPFCM is based on the previous two mathematical theories, the theory of fuzzy sets and possibility theory. Our contribution to hybrid HPFCM uses the results of the FCM algorithm as input data for the PCM algorithm. The degree of membership matrix PCM is initialized by the value of the result matrix of FCM. We present our algorithm in Algorithm 4.

Let 𝑋={𝑋𝑖/𝑖={1⋯𝑁}} the data sets, 𝑀: the maximum number of iterations, 𝐾: the number of the clusters, 𝑈: the matrix of membership degrees function, 𝑉: the matrix of cluster center, 𝑚: the fuzzy degree, Fuzzy_classification(): is the function of fuzzy classification, F_obj(): the objective function, Possibilist_classification(): the possibilistic classification function, 𝜂𝑗: Degree of weight possibilistic, and 𝜖: the threshold Representing the convergence error. (1) Step 1. Initialize the parameters (2)  (i) 𝑚=1.5 (3)  (ii) Initialize the matrix 𝑉 by center clusters (4)  (iii) Initialize the matrix 𝑈 by membership degrees, random values in the interval [0,1] and it also satisfies the condition in (3). (5) Step 2. Fuzzy process (6) Repeat (7)  𝑉old=𝑉;𝑈old=𝑈 (8)  Compute: [𝑈,𝑉]= Fuzzy_classification (𝑋,𝑈,𝑉,𝑚) // with (4) and (5) (9)  𝑉new=𝑉;𝑈new=𝑈 (10) Until 𝑉new−𝑉old≤𝜀 (11) Step 3. Possibilist process (12)  𝑊=𝑈new // Initialize the matrix 𝑊 by 𝑈new (13)  𝑌=𝑉new // Initialize the matrix 𝑌 by 𝑉new (14) Repeat (15)  𝑌old=𝑌;𝑊old=𝑊 (16)  Compute: [𝑊,𝑌]=  Possibilist_classification (𝑋,𝑊,𝑌,𝜂) // with (7) and (8) (17)  𝑌new=𝑌;𝑊new=𝑊 (18) Until 𝑌new−𝑌old≤𝜀

Algorithm 4 

HPFCM.

4. Experimental Results

4.1. Synthetic Data Sets

Data Set 𝑋₁₂. This data set consists of 10 patterns and two outliers (noise), as shown in Table 1 and as given in [30]. We applied the FCM, PCM, PFCM, and HPFCM to 𝑋₁₂ with initial parameters 𝑚=2, 𝐾=2 (number of clusters), and 𝜂=2 and the center matrix 𝑉𝑣₁𝑣₂ with random values. The ideal (true) centroids are 𝑉_ideal=−3.1403.140. (13)

Table 1 

Data sets 𝑋₁₂.

Table 2 shows the results of prototypes of center clusters 𝑣₁ and 𝑣₂ using the four algorithms. For calculating the error between the results prototypes and ideal center clusters, we used the formula: Er=‖𝑉_ideal−𝑉_∗‖², where * is the FCM, PCM, PFCM, and HPFCM. So, we can have the errors as follows: Er_fcm=0.414, Er_pcm=1.416, Er_pfcm=0.3796, and Er_hpfcm=0.0488.

Table 2 

Prototypes results of 𝑉, produced by FCM, PCM, PFCM, and HPFCM with data set 𝑋₁₂.

These error values show that our method gives the best prototypes for 𝑣₁ and 𝑣₂. And Figure 1 shows the effectiveness of our approach.

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Figure 1 

Clustering results of four algorithms FCM, PCM, PFCM, and HPFCM with data set 𝑋12.

4.2. Real Data Sets of UCI

4.2.1. Description of UCI Data Sets

In this section, we tested the performance of our method in different real data sets: Iris, Glass, Wine, and Wisconsin Breast cancer, as shown in Table 3. These data sets are referenced in Benchmark of UCI Repository of Machine Learning Data base [31].

Table 3 

Characteristics of data sets: Iris, Wine, Breast-w, and Glass. Ref. data sets UCI [31].

Iris Data Set. It describes a type of Iris plant, and it contains 3 classes of 50 instances for each class. There is one class (class 1) which is linearly separable from the other, but the classes 2 and 3 are not linearly separable, and it is a four-dimensional data set [31].

Wine Data Set. The Wine is a 13-dimensional data set and contains the values of a chemical analysis of Wines grown in the same region in Italy but derived from three different cultivars [31].

Glass Data Set. Glass is a 9-dimensional data set that contains 214 objects of 6 classes representing the types of glass [31].

Breast Cancer Data Set. The Wisconsin Breast cancer is a 9-dimensional data set that contains 699 objects of 2 classes representing the types of benign and malignant cancers [31].

4.2.2. Clustering Results for UCI Data Sets

To evaluate the clustering accuracy, we used the index of Huang and Ng [32] as given in ∑ℎ=^𝐾𝑖=1𝑐_𝑖𝑛, (14) where 𝐾 is the number of clusters, 𝑐_𝑖 is the number of patterns occurring in both cluster 𝑖th and its true corresponding, and 𝑛 is the total number of objects in the data set.

In Table 4, using Iris data set, we can see that PCM algorithm gave less accuracy (0.667) than other algorithms, and the algorithms FCM and PFCM gave the values 0.893 and 0.9, respectively, and our approach gave the best accuracy (0.929). And using other data sets, Wine, Glass, and Breast, we can also see that all accuracy values of our approach are better compared to other algorithms: FCM, PCM, and PFCM.

Table 4 

Clustering accuracy by index Huang for FCM, PCM, PFCM, and HPFCM with four data sets: Iris, Wine, Glass, and Breast cancer.

4.3. Medical MR Images Data Sets

4.3.1. Validity Index

There are several validity indexes to evaluate the performance of a clustering method [25]. In our work, we used the sensitivity index (Se), specificity (Sp), and the classification accuracy (AC). These three indexes are based on the four parameters: true positive (TP), true negative (TN), false positive (FP), and false negative (FN), shown in Figure 2, Tables 5 and 6.(i)TP is a true positive, when a pixel labeled is correctly classified as positive.(ii)TN is a true negative, when a pixel labeled is correctly classified as negative.(iii)FP is a false positive, when a negative labeled pixel is incorrectly classified as positive.(iv)FN is a false negative, when a positive labeled pixel is incorrectly classified as negative.

Table 5 

Index description.

Table 6 

Regions description.

Figure 2 

Description of the sets.

Sensitivity (Se) index gives the proportion of true positives over all structures that should be segmented Se=TP.TP+FN (15)

Specificity (Sp) index gives the proportion of true negatives over all structures, that should not be segmented Sp=TN.TN+FP (16)

Classification accuracy (CA) index gives the performance of a method: CA=TP+TN.TP+FN+TN+FP (17)

4.3.2. Brainweb: Simulated Brain Database

This data set can simulate the brain MR images in three orthogonal views (transversal, sagittal, and coronal); it can also give three sequences volumes (T1, T2, and PD (proton density)), as shown in Figures 3, 4, and 5 [41], and a variety of slice thicknesses, noise levels, and levels of intensity nonuniformity [41].

Figure 3 

Brain MR image 3D.

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Figure 4 

Brain MR images with three views axial, coronal, and sagittal [41].

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Figure 5 

Brain MR images with three sequences volumes T1, T2, and DP [41].

Tables 7, 8, and 9 show the clustering results using the different brain tissues: gray matter (GM), white matter (WM), and cerebrospinal fluid (CSF), respectively. We used the different noise levels (0%, 1%, 3%, 5%, 7%, and 9%), in order to evaluate our approach HPFCM and we compared the accuracy results with other algorithms, FCM, PCM, and PFCM. We can see that FCM algorithm is not stable and very sensitive to noises, and PCM algorithm gave stable values, but it did not show enough performance; instead PFCM and HPFCM gave good performance and were stable. Also, we plotted the curves for visualizing the results, as can be seen in Figure 6, that our approach gave the best accuracy compared to PFCM and other conventional algorithms FCM and PCM. The clustering results for brain MR image with Ground Truth are shown in Figure 7.

Table 7 

Results comparison for clustering index (Se and Sp) for GM tissue by FCM, PCM, PFCM, and HPFCM, with different noises (0%, 1%, 3%, 5%, 7%, and 9%), using Brainweb data sets.

Table 8 

Results comparison for clustering index (Se and Sp) for WM tissue by FCM, PCM, PFCM, and HPFCM with different noises (0%, 1%, 3%, 5%, 7%, and 9%), using Brainweb data sets.

Table 9 

Results comparison for clustering index (Se and Sp) for CSF tissue by FCM, PCM, PFCM, and HPFCM, with different noises (0%, 1%, 3%, 5%, 7%, and 9%), using Brainweb data sets.

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Figure 6 

Curves of clustering index (Se and Sp) for CSF, WM, and GM, using four algorithms, FCM, PCM, PFCM, and HPFCM.

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Figure 7 

Clustering results using four algorithms: (a) original image, (b) Ground Truth, (c) FCM, (d) PCM, (e) PFCM, and (f) HPFCM, using brain MR images from Brainweb.

4.3.3. IBSR Data Set

To evaluate our approach on real medical images, we used the IBSR data sets (Internet Brain Segmentation Repository), which is given by the Center for Morphometric Analysis of Massachusetts General Hospital [42].

Table 10 shows the clustering results of different brain tissues: GM, WM, and CSF. We did the study on 10 images from data sets IBSR; each image has 150 × 256 × 256 voxels. Also, the Ground Truth was established by the experts and it is available in IBSR [42].

Table 10 

Clustering by FCM, PCM, PFCM, and HPFCM, using IBSR dataset.

According to the results of Table 11, we can see that our approach HPFCM gave the best results using different indexes: specificity (Se), sensitivity (Sp), and clustering accuracy (CA). In order to better analyze the results, we established the curve, as shown in Figure 8, so this curve shows that our approach is a better model and very efficient compared to other algorithms FCM, PCM, and PFCM. The clustering results of brain MR images of IBSR with Ground Truth are shown in Figure 9.

Table 11 

Accuracy clustering by FCM, PCM, PFCM, and HPFCM, using IBSR dataset.

Figure 8 

Results of clustering accuracy for the algorithms, FCM, PCM, PFCM, and HPFCM, using MR images from IBSR.

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Figure 9 

Clustering results using four algorithms: (a) original image, (b) Ground Truth, (c) FCM, (d) PCM, (e) PFCM, and (f) HPFCM, using brain MR images from IBSR data sets.

5. Conclusion

In this paper, we presented a new approach to clustering, which is based on the fuzzy c-means and possibilistic c-means. Our approach is used to model uncertain and imprecise data, in order to segment the brain tissue for medical images MRI. And it has been compared to three conventional clustering algorithms: FCM, PCM, and PFCM. The performance of our approach has been successfully validated on several synthetics data sets and real medical images MRI. In the future, we will integrate other theories: the genetic and evidence theory, in order to optimize our hybrid method and we can have a very robust modeling for complex data.